python
import sys
from bisect import bisect_left, bisect_right


class FenwickMax:
    """Fenwick tree supporting point updates and prefix maximum queries."""

    def __init__(self, size: int):
        self.size = size
        self.tree = [0] * (size + 1)

    def update(self, index: int, value: int) -> None:
        index += 1  # Convert to 1-based indexing

        while index <= self.size:
            if value > self.tree[index]:
                self.tree[index] = value
            index += index & -index

    def query(self, count: int) -> int:
        """Maximum among indices [0, count)."""
        result = 0

        while count > 0:
            result = max(result, self.tree[count])
            count -= count & -count

        return result


def solve(N: int, M: int, P: list[int]) -> int:
    if N == 0:
        return 0

    # Coordinate compression of prices.
    values = sorted(set(P))
    size = len(values)

    # States:
    # 0: last step was UP,   mulligan unused
    # 1: last step was DOWN, mulligan unused
    # 2: last step was UP,   mulligan used
    # 3: last step was DOWN, mulligan used
    prefix = [FenwickMax(size) for _ in range(4)]
    suffix = [FenwickMax(size) for _ in range(4)]

    answer = 1

    for price in P:
        index = bisect_left(values, price)

        # Previous prices eligible for an UP step:
        # previous_price <= price - M and previous_price < price.
        lower_count = min(
            bisect_right(values, price - M),
            bisect_left(values, price)
        )

        # Previous prices eligible for a DOWN step:
        # previous_price >= price + M and previous_price > price.
        higher_start = max(
            bisect_left(values, price + M),
            bisect_right(values, price)
        )

        def lower_query(state: int) -> int:
            return prefix[state].query(lower_count)

        def higher_query(state: int) -> int:
            # Prices are reversed in the suffix Fenwick trees.
            return suffix[state].query(size - higher_start)

        # Current step is UP and alternates after a DOWN step.
        previous_down_unused = lower_query(1)
        up_unused = (
            previous_down_unused + 1
            if previous_down_unused > 0
            else 1
        )

        # Current step is DOWN and alternates after an UP step.
        previous_up_unused = higher_query(0)
        down_unused = (
            previous_up_unused + 1
            if previous_up_unused > 0
            else 1
        )

        # End with an UP step after using the mulligan:
        # 1. Alternate after a DOWN step where mulligan was already used.
        # 2. Follow another UP step and use the mulligan now.
        previous_down_used = lower_query(3)
        previous_up_same = lower_query(0)

        up_used = max(
            previous_down_used + 1 if previous_down_used > 0 else 0,
            previous_up_same + 1 if previous_up_same > 0 else 0
        )

        # End with a DOWN step after using the mulligan:
        # 1. Alternate after an UP step where mulligan was already used.
        # 2. Follow another DOWN step and use the mulligan now.
        previous_up_used = higher_query(2)
        previous_down_same = higher_query(1)

        down_used = max(
            previous_up_used + 1 if previous_up_used > 0 else 0,
            previous_down_same + 1 if previous_down_same > 0 else 0
        )

        current = [
            up_unused,
            down_unused,
            up_used,
            down_used
        ]

        # Insert the current price after calculating all states, so an
        # element cannot transition to itself.
        reversed_index = size - 1 - index

        for state, length in enumerate(current):
            if length > 0:
                prefix[state].update(index, length)
                suffix[state].update(reversed_index, length)

        answer = max(answer, *current)

    return answer


def main() -> None:
    data = list(map(int, sys.stdin.buffer.read().split()))

    if len(data) < 2:
        return

    N = data[0]
    M = data[1]
    P = data[2:2 + N]

    print(solve(N, M, P))


if __name__ == "__main__":
    main()

The four dynamic-programming states track the last direction and whether the mulligan has been used. Fenwick trees retrieve the best earlier subsequence whose final price is at least M below or above the current price.

Complexity: O(N log N) time and O(N) space.