Existence and uniqueness conjecture for two polynomial difference equations

Let $u$ be a nonnegative integer and let $P\in\mathbb{Q}[x]$ be a polynomial. Consider the two polynomial difference equations

$$(x+2u+4)(x+u+1)P(x+1)-(x-1)(x+u-1)P(x-1) =((2u^2+8u+7)x+(2u+3)(u^2+3u+1))P(x).$$ $$(x+2u+4)(x+u+4)P(x+1)-(x-1)(x+u+2)P(x-1) =((2u^2+8u+7)x+(2u+3)(u+2)(u+3))P(x).$$

Polynomial difference-equation conjecture. For each nonnegative integer $u$, the first equation has, up to multiplication by a nonzero constant, a unique polynomial solution $P$, and this solution satisfies $\deg(P)=u(u+3)$. The same assertion holds for the second equation.

These polynomial solutions are used in the continued-fraction construction of the paper's one-parameter families associated with $\pi^2$. The conjecture is presented as an unproved existence, uniqueness, and degree assertion; the supplied text gives no resolution.