Uniform error conjecture for symmetric integer matrices with prescribed row sums

Let $M(n,\ell)$ denote the number of symmetric $n\times n$ matrices with entries in $\{0,1,\ldots,\ell\}$ and every row sum equal to $\ell$, and let $M_{\mathrm{naive}}(n,\ell)$ be the naive asymptotic estimate defined earlier in the paper. For even $n\ell$, define $\varDelta(n,\ell)$ by

$$M(n,\ell)=M_{\mathrm{naive}}(n,\ell)\,\sqrt{2}\exp\biggl(\frac{3}{4}+\frac{3\ell+1}{12\ell(n-1)}+\frac{\varDelta(n,\ell)}{n(n-1)}\biggr).$$

Uniform error conjecture. For $n\geq 5$ and $\ell\geq 1$, one has

$$\lvert\varDelta(n,\ell)\rvert<1.$$

This conjecture asserts that the asymptotic expansion has a substantially smaller error than the authors can prove and gives a uniform bound on the remaining correction term for all admissible even pairs $(n,\ell)$.