Generalized lattice-path trigonometric positivity conjecture

Let $M_i$ and $N_i$ be nonnegative integers such that $|M_i-N_i|\leq k$ for $i=1,\ldots,r$. Define the sums

$$\sum_l\prod_{i=1}^r {M_i+N_i\choose M_i-kl}\cos(lx)$$

and

$$\sum_l\prod_{i=1}^r {M_i+N_i\choose M_i-kl}\frac{\sin(|l|x)}{\sin(x)}.$$

Generalized positivity conjecture. The first sum is a polynomial in $1+\cos(x)$ with nonnegative integral coefficients, and the second satisfies

$$\sum_l\prod_{i=1}^r {M_i+N_i\choose M_i-kl}\frac{\sin(|l|x)}{\sin(x)}\geq 0$$

for any real $x$.

This conjecture proposes a simultaneous extension of the preceding two-factor positivity results to an arbitrary number $r$ of pairs $(M_i,N_i)$. The supplied text gives no resolution or further evidence beyond the statement, so its status remains open.