The convex-hull conjecture for the Mathieu conjecture on $SU(2)$

Let $f=\sum_{i=1}^{k}A_i t^{\ell_i}_{m_i,n_i}$ be an $SU(2)$-finite function, with $A_i\neq0$ for every $1\leq i\leq k$. Let $C$ be the closed convex hull of the points $\bigl((m_i,n_i)\bigr)_{i=1}^k$. The convex-hull conjecture. The following are equivalent:

$$\int_{SU(2)}f(g)^P\,dg=0\quad\text{for all }P\in\mathbb{N}_{>0}$$

and

$$(0,0)\notin C.$$

This is proposed as an alternative formulation for the Mathieu conjecture for $SU(2)$. The preceding discussion explains that the equivalence is clear except for possible cancellations in the rank-two case, and the source gives no resolution.