A Bernstein-type inequality for translates of powers of a cosine bump

Let $(\lambda_n)_{n=-\infty}^{\infty}$ be a strictly increasing sequence of real numbers. Define

$$H(x)= \begin{cases} \cos x,& |x|\le \pi/2,\\ 0,& |x|\ge \pi/2. \end{cases}$$

For a positive integer $M$, suppose that the sequence satisfies the gap condition

$$\lambda_{n+M}-\lambda_n\ge \pi$$

for all $n$. For every finite sequence $(a_n)$ of real numbers, set

$$G(x)=\sum a_n H^M(x+\lambda_n).$$

Bernstein-type conjecture. The function $G$ satisfies

$$\int_{-\infty}^{\infty}|G'(x)|^2\,dx\le M^2\int_{-\infty}^{\infty}|G(x)|^2\,dx.$$

This conjecture generalizes the preceding inequality for a single power $H^M$, which follows from Bernstein's inequality for trigonometric polynomials via the substitution $T(x)=\sin^M x$. The claim asks whether the same $L^2$ derivative bound remains valid for arbitrary finite linear combinations of translates whose locations obey the stated gap condition; the supplied text gives no resolution.