The trisection classification conjecture for the standard four-sphere

Let $\mathcal{T}$ be a trisection of the standard smooth four-sphere $\mathcal{S}^4$. Let $\mathcal{S}^0$ denote its standard $(0,0)$-trisection, and let $\mathcal{S}^{k_1,k_2,k_3}$ denote the trisection obtained from $\mathcal{S}^0$ by performing $k_i$ $i$-stabilizations for $1\leq i\leq 3$. Trisection classification conjecture. Every trisection $\mathcal{T}$ of $\mathcal{S}^4$ is isotopic to $\mathcal{S}^0$ or one of its stabilizations $\mathcal{S}^{k_1,k_2,k_3}$. This is proposed as a four-dimensional analogue of Waldhausen's theorem for $S^4$; the source notes that it is likely false, and gives no resolution here.