Conjecture on log-concavity and unimodality of involution statistics

For $n\geq 4$, let $\mathcal{I}_n^{\mathrm{maj}}(q)$ and $\mathcal{I}_n^{\mathrm{inv}}(q)$ denote the major-index and inversion generating polynomials over involutions, and let $Inv_n^B(q)$, $Inv_n^D(q)$, and $Inv_n^O(q)$ be the corresponding inversion polynomials for the types named in the source. For a polynomial $P(q)$, write $[q^i]P(q)$ for the coefficient of $q^i$.

The conjecture. For all $n\geq 4$:

1. The sequence $\{[q^i]\mathcal{I}_n^{\mathrm{maj}}(q)\}_{i=0}^{\binom{n}{2}}$ is log-concave.
2. For $2\leq i\leq \binom{n}{2}-2$,

$$\bigl([q^i]\mathcal{I}_n^{\mathrm{inv}}(q)\bigr)^2\geq\bigl([q^{i-2}]\mathcal{I}_n^{\mathrm{inv}}(q)\bigr)\bigl([q^{i+2}]\mathcal{I}_n^{\mathrm{inv}}(q)\bigr).$$

3. The sequences $\{[q^{2i}]Inv_n^B(q)\}_{i\geq0}$ and $\{[q^{2i+1}]Inv_n^B(q)\}_{i\geq0}$ are unimodal.
4. The sequences $\{[q^{2i}]Inv_n^D(q)\}_{i\geq0}$ and $\{[q^{2i+1}]Inv_n^D(q)\}_{i\geq0}$ are unimodal.
5. The sequences $\{[q^{2i}]Inv_n^O(q)\}_{i\geq0}$ and $\{[q^{2i+1}]Inv_n^O(q)\}_{i\geq0}$ are unimodal.

The conjecture extends the original log-concavity conjecture. The supplied passage reports numerical log-concavity through $n\leq14$ and notes that the relevant roots are not real and do not lie in the stated triangular region, but gives no resolution of these assertions.