Three-way desingularization conjecture for associative $T^2$-cone singularities

Let $(\psi_t)_{t\in(-T,T)}$ be a $1$-parameter family of closed, tamed, definite $4$-forms on $Y$. Let $\hat P$ be an unobstructed singular associative submanifold in $(Y,\psi_0)$ with a unique singularity at $x$ modeled on $\hat L$. Let $\delta_1,\delta_2,\gamma\in\mathbf R$ be the associated constants. Three-way desingularization conjecture. For a generic family, if $\delta_1\neq0$, $\delta_2\neq0$, $\delta_1\neq\delta_2$, and $\gamma\neq0$, then for sufficiently small $\varepsilon>0$ there are three unobstructed associative immersions $\iota^i_\varepsilon:P^i\to Y$, $i=1,2,3$, at parameters $t_i(\varepsilon)$, each close to $\hat P$ away from $x$ and to the corresponding local model $L^i_\varepsilon$ near $x$. Their domains are obtained by Dehn filling $\hat P^\circ=\hat P{\backslash} B_\sigma(x)$ along the slopes $\mu_i=(0,1),(-1,0),(1,-1)$, and

$$t_1(\varepsilon)=-\frac{\delta_2}{\gamma}\varepsilon+O(\varepsilon^2),\qquad t_2(\varepsilon)=\frac{\delta_1}{\gamma}\varepsilon+O(\varepsilon^2),\qquad t_3(\varepsilon)=\frac{\delta_2-\delta_1}{\gamma}\varepsilon+O(\varepsilon^2).$$

This predicts three possible smoothings of the singular associative and describes the corresponding first-order parameter shifts in a generic one-parameter deformation. It is intended to explain the wall-crossing transition associated with an isolated associative singularity modeled on the cone over $T^2$.