The chain-homotopy conjecture for obstruction complexes of model fibers

Let $(X,T^*_{X,-})$ be a model and let $(Y,T^*_{Y,-})$ be a $gt$-model over a connected Stein base whose fiber is $(X,T^*_{X,-})$, so that $Y=X\times B$ and $T^*_{Y,-}=p_X^*T^*_{X,-}$. Write $\operatorname{Ob}(X,T^*_{X,-})$ and $\operatorname{Ob}(Y,p_X^*T^*_{X,-})$ for their obstruction complexes. Chain-homotopy conjecture. The obstruction complexes

$$\operatorname{Ob}(X,T^*_{X,-})\quad\text{and}\quad\operatorname{Ob}(Y,p_X^*T^*_{X,-})$$

are chain homotopic. The conjecture is motivated by the asserted constancy of the obstruction class to splitting the fiber of a $gtm$-family with this total-space model. Its resolution status is not specified in the supplied text.