Alternating bi-3-positivity conjecture for BorosMoll polynomials

For $m1$, define the BorosMoll polynomial $M_m(x)$ and its transform

$$Q_m(x)=2^m m!x^mM_m\left(\frac{1}{x}\right).$$

A polynomial is symmetric if its coefficients read the same forwards and backwards, unimodal if its coefficients increase and then decrease, and alternatingly bi-\gamma-positive in the sense used for these polynomials. BorosMoll positivity conjecture. For every $m1$, there are symmetric and unimodal polynomials $a_m(x)$ and $b_m(x)$ such that

$$Q_m(x)=a_m(x)+xb_m(x),$$

and $Q_m(x)$ is alternatingly bi-\gamma-positive. Consequently, $M_m(x)$ is also alternatingly bi-\gamma-positive. This proposes a refined positivity and unimodality structure for the BorosMoll polynomials; the supplied text gives empirical evidence but no resolution.