Connected sums preserve algebraic overtwistedness

Let $(M,\xi)$ be a cooriented contact manifold. A contact manifold $(N,\eta)$ is algebraically overtwisted when its contact homology vanishes, $HC_*(N,\eta)=0$.

Connected-sum conjecture. If $(N,\eta)$ is algebraically overtwisted, then

$$(M,\xi)\#(N,\eta)$$

is also algebraically overtwisted.

This conjecture predicts that vanishing contact homology is preserved after taking a contact connected sum with an arbitrary cooriented contact manifold. The paper states it as evidence-based and does not provide a resolution.