The nonnegative invariant-polynomial conjecture in dimension three

For $d\geq 6$, define

$$Q_d(a_1,a_2)=-\frac{T_d\left((a_1+a_2)/\sqrt{2},(a_1-a_2)/\sqrt{2}\right)}{a_1^2-a_2^2}.$$

Here $\mathbb{R}_{\geq0}[a_1^2+a_2^2,a_1^2a_2^2]$ denotes the polynomials in the two displayed generators with nonnegative real coefficients. Nonnegative invariant-polynomial conjecture. One has

$$Q_d(a_1,a_2)\in\mathbb{R}_{\geq0}[a_1^2+a_2^2,a_1^2a_2^2].$$

This is stated as a strengthening of the type-$D$ sign conjecture for $n=3$, replacing a sign prediction by coefficientwise nonnegativity in the relevant invariant variables. The source does not provide a resolution.