Here are a few more ways to rephrase the problem. Let $A_k$ denote a $kn\times kn$ matrix, composed of $k\times k$ blocks, where the $ij$ block is filled with copies of a standard exponential random variable $\gamma_{ij}$. The $\gamma_{ij}$'s are taken to be independent. A. Barvinok showed that $$M(n,k)=\frac{\mathbb{E}(\operatorname{per}(A_k))}{(k!)^{2n}}$$ and used this to prove that $M(n,k)$ is almost log-concave in $k$. More specifically, he showed in "Brunn-Minkowski inequalities for contingency tables and integer flows", Adv. in Math., 211 (2007), 105-122, that the following inequality holds $$\alpha(k)M(n,k)^2\geq M(n,k-1)M(n,k+1)$$ with $\alpha(k)=O(k^n)$. He conjectures that this inequality holds for $\alpha=1$, but as far as I know, this remains open. I'm not sure what's the best upper bound for $$\frac{M(n,k)M(n,n+k+1)}{M(n,k+1)M(n,k+n)}$$ that one gets using this analytic approach.

Another way of stating the problem is through RSK, and turn it into an inequality in terms of Kostka numbers. If $\tau(k)$ stands for the partition $(k,k,\dots,k)$, with $k$ parts. Then log-concavity is equivalent to $$\left(\sum_{\lambda} K_{\lambda \tau(k)}\right)^2\geq \left(\sum_{\lambda} K_{\lambda \tau(k-1)}\right)\left(\sum_{\lambda} K_{\tau(k+1)}\right)$$ and your inequality can be written similarly. One might hope that there is a proof making use of known inequalities between Kostka numbers or known Schur positivity results.

Yet another equivalent formulation comes from Pietro's answer. We need to prove that the coefficient of $(x_1y_1\cdots x_ny_n)^k(z_1t_1\cdots z_nt_n)^{k+1}$ is non-negative in $$\frac{\left(z_1t_1\cdots z_nt_n-x_1y_1\cdots x_ny_n\right)}{\prod_{i,j=1}^n(1-x_iy_j)(1-t_iz_j)}.$$

In a different direction, we have $$M(n,k)=\sum_{i=0}^d h_i\binom{k+i}{d}$$ where $d=(n-1)^2$. In "Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley", C.A. Athanasiadis proved that $(h_0,h_1-h_0,\dots,h_{\lfloor d/2\rfloor}-h_{\lfloor d/2\rfloor-1})$ is a g-vector, so it satisfies the inequalities of Mcmullen's g-theorem. In particular $h$ is a symmetric unimodal sequence. I doubt that this is enough to conclude log-concavity of $M(n,k)$, or your property for that matter, but perhaps it gives some insight on how hard the problem is.