The inclusion conjecture for decomposable clutters

Let $4\mathfrak{C}_d4$ denote the class of chordal $d$-uniform clutters and let $4\mathfrak{C}'_d4$ denote the class of decomposable $d$-uniform clutters. The inclusion conjecture.

$$\mathfrak{C}'_d \subset \mathfrak{C}_d,$$

for all $d$. For $d=2$, the inclusion is known, in fact equality follows from Dirac's theorem. The conjecture is motivated by results on pure skeleta of quasi-forests, stable and lexsegment ideals, and experimental evidence; its status is not resolved in the supplied text.