The conjecture characterizing graphs with fixed PSD propagation time one

Let $G$ be a connected graph. A PSD fast join is a graph of order $n\geq 2$ such that $G\cong K_n$ or

$$G\cong (K_{n_1}\mathbin{\,\dot{\cup}\,} K_{m_1})\vee \ldots \vee (K_{n_t}\mathbin{\,\dot{\cup}\,} K_{m_t}),$$

for $t\geq 2$ and positive integers $n_i,m_i$ with $n=\sum_{i=1}^t n_i+m_i$. Here $\overline{\operatorname{pt}_+}(G)$ denotes the maximum positive semidefinite propagation time over minimum positive semidefinite forcing sets. PSD fixed-propagation-time conjecture. If $G$ is a connected graph and

$$\overline{\operatorname{pt}_+}(G)=1,$$

then $G$ is a PSD fast join. The theorem establishing the converse shows that PSD fast joins have fixed PSD propagation time equal to one; the conjecture is proved for graphs that are joins or have sufficiently large forcing numbers, while the general case remains open.