Monotonicity conjecture for extremal depth-one quasimodular forms

Let $w$ be an even integer with $w \ge 6$ and $w \ne 8,10$, and let $X_{w,1}$ denote the extremal quasimodular form of weight $w$ and depth $1$. Monotonicity conjecture. For all even $w \ge 6$ but $w = 8, 10$, the function

$$t \mapsto t^{w-1} X_{w,1}(it)$$

is monotone decreasing for $t > 0$. This extends the monotonicity properties proved for several specific weights, but the general assertion remains open.