Square-placement Wilf-equivalence conjecture for three length-four patterns

For a distant pattern, write a square between two consecutive letters to indicate the constrained gap at that position. Consider the classical patterns $1234$, $1243$, and $2143$, and choose the same one of the three positions between consecutive letters in each pattern. Square-placement Wilf-equivalence conjecture. The resulting three distant patterns are Wilf-equivalent; for example,

$$|\operatorname{Av}_{n}(1\square 234)|=|\operatorname{Av}_{n}(1\square 243)|=|\operatorname{Av}_{n}(2\square 143)|.$$

The paper calls this conjecture surprising and notes that an analogous assertion for arbitrary Wilf-equivalent classical patterns is false, giving a counterexample in length four. It remains open in the stated special case.