Simple angular dependence conjecture for fundamental free-plate modes

Let $\mathbb{B}(R)$ be a ball of radius $R$, and consider the free-plate eigenvalue problem with tension $\tau>0$ and Poisson's ratio $\sigma\in[0,1)$. Let $\omega_1(\tau,\sigma)$ be its fundamental eigenvalue, and let $M$ denote the boundary operator appearing in the free boundary condition. For positive constants $a$ and $b$ and a real constant $\gamma$, consider

$$u_1(r,\hat{\theta})=\left(j_1(ar/R)+\gamma i_1(br/R)\right)Y_1(\hat{\theta}),$$

where $a$ and $b$ satisfy

$$b^2-a^2=R^2\tau,\qquad a^2b^2=R^4\omega_1(\tau,\sigma),$$

and

$$\gamma=\frac{-Mj_1(a)}{Mi_1(b)}.$$

Simple angular dependence conjecture. The fundamental modes of $\mathbb{B}(R)$ can be written as linear combinations of $u_1(r,\hat{\theta})$.

This conjecture would establish the simple angular dependence needed for the paper's isoperimetric argument and hence support the inequality for all plates. It is motivated by the known $\sigma=0$ case and additional numerical and analytic evidence; the general claim remains open.