Conjecture on irrelevant pathhood of interpreted ParamDTT types

Let $T$ be a type that can be constructed in ParamDTT. Say that $T$ has irrelevant pathhood when weakening paths to bridges is injective:

$$(W,\mathbf{i}:\mathbb{P})\mathrel{\rhd}t:T\left[\gamma\right\rangle\mapsto(W,\mathbf{i}:\mathbb{B})\mathrel{\rhd}t\angles{\mathbf{i}/\mathbf{i}}:T\left[\gamma(\mathbf{i}/\mathbf{i})\right\rangle.$$

Pathhood-irrelevance conjecture. The interpretation of any type that can be constructed in ParamDTT has irrelevant pathhood. This property is introduced to ensure injectivity of the operation sending $t:T$ to $t[\kappa]:T[\kappa]$. The source gives no proof or resolution of the conjecture.