Characterization conjecture for forcing number and parallel paths

Let $G$ be a graph with maximum degree at most $3$, let $n$ be the number of vertices of $G$, and let $F(G)$ denote its forcing number. A $k$-parallel path graph is a graph of $k$-parallel paths. Parallel-path characterization conjecture. For some $k\in[n]$,

$$F(G)=k \quad\Longleftrightarrow\quad G\text{ is a graph of }k\text{-parallel paths}.$$

The conjecture seeks to characterize all forcing numbers of graphs of maximum degree at most three by the number of parallel paths. The source says it is supported by earlier results and observations, but gives no resolution status.