Higher Apéry-limit conjecture for sums of powers of binomial coefficients

Let

$$A^{(d)}(n)=\sum_{k=0}^n\binom{n}{k}^d$$

be the sum of the $d$th powers of the binomial coefficients. Let the minimal-order recurrence satisfied by $A^{(d)}(n)$ have solutions normalized by $C^{(d)}(0)=0$ and $C^{(d)}(1)=1$.

Higher Apéry-limit conjecture. For $d\geq 5$, there is a unique such solution $C^{(d)}(n)$ satisfying

$$\lim_{n\to\infty}\frac{C^{(d)}(n)}{A^{(d)}(n)}=\frac{3(5d+2)}{(d+1)(d+2)(d+3)}\zeta(4).$$

The formula is supported by numerical evidence for $5\leq d\leq 10$, where the corresponding limits were computed to high precision. No general proof is given in the source, so the conjecture remains open.