Crouzeix's conjecture for scaled q-numerical ranges

Let $A$ be an $n\times n$ matrix, let $0<|q|\leq 1$, and let $\Omega_q$ denote the scaled $q$-numerical range introduced in the source. For a polynomial $p$ with complex coefficients, the scaled $q$-numerical-range conjecture.

$$\|p(A)\|\leq\max\!\left(1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right)\sup_{z\in\Omega_q}|p(z)|.$$

This conjecture generalises Crouzeix's conjecture from the classical numerical range to scaled $q$-numerical ranges. When $q=1$, the set $\Omega_q$ reduces to the numerical range $W(A)$, recovering the known $1+\sqrt{2}$ spectral-constant bound; the proposed inequality is presented without a resolution in the source.