Polynomial-time solvability conjecture for MOTJ and SOC

Let MOTJ denote the problem of deciding, for an undirected graph $G=(V,E)$ with rational edge weights $w:E\to\mathbb{Q}$, a set $T\subseteq V$, and $k\in\mathbb{Q}$, whether there is an odd $T$-join of total weight at most $k$. Let SOC denote the corresponding problem of finding a minimum-weight odd cycle. Polynomial-time solvability conjecture. MOTJ and SOC can be solved in polynomial time. A randomized polynomial-time algorithm places MOTJ in the class $\operatorname{RP}$, so polynomial-time solvability remains conjectural; the source presents these problems as potentially more tractable than the shortest odd path problem.