Conjectural inversion enumerations for Magog matrices

Let $\mathrm{Magog}_n$ denote the set of Magog matrices of size $n$, and let $\mathrm{posinv}$ and $\mathrm{inv}$ denote the positive inversion statistic and inversion statistic, respectively. For a statistic $s$, write $|\mathrm{Magog}_n(k~s)|$ for the number of matrices in $\mathrm{Magog}_n$ with statistic $s$ equal to $k$. The conjectured enumerations are:

Inversion-enumeration conjecture.

$$|\mathrm{Magog}_n(1~\mathrm{posinv})|=\binom{n}{2},\qquad n\geq 2,$$ $$|\mathrm{Magog}_n(2~\mathrm{posinv})|=2\binom{n-1}{2}+4\binom{n-1}{3}+3\binom{n-1}{4},\qquad n\geq 3,$$ $$|\mathrm{Magog}_n(\binom{n}{2}-2~\mathrm{posinv})|=n(n-2),\qquad n\geq 3,$$ $$|\mathrm{Magog}_n(\binom{n}{2}-1~\mathrm{inv})|=2^n-n-1,$$

the $n$th Eulerian number. These formulas are based on numerical data; the second sequence matches OEIS sequence A090809, which counts the coefficient of a certain irreducible character of the symmetric group in a certain Kronecker power. Further verification or proofs of the enumerations remain open.