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 $B^{(d)}(0)=0$ and $B^{(d)}(1)=1$.

Apéry-limit conjecture. For $d\geq 3$, there is a unique such solution $B^{(d)}(n)$ satisfying

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

The cases $d=3$ and $d=4$ were numerically observed, and the case $d=3$ was proved by Zagier using modular forms. The conjecture is numerically confirmed through $d=10$, while the general assertion remains open.