Categorical compactness conjecture for conically smooth stratified spaces

Let $(X,P)$ be a conically smooth stratified space. Say that $(X,P)$ is of finite stratified type and categorically compact in the senses defined in the paper.

Categorical compactness conjecture. If $(X,P)$ is of finite stratified type, then $(X,P)$ is also categorically compact.

This conjecture would extend Volpe's theorem by enlarging the class of stratified spaces known to have the relevant categorical compactness property. The paper states that it is not known whether every conically smooth stratified space of finite stratified type is automatically finitary, and provides no resolution of this conjecture.