The nonexistence conjecture for M-property incidence graphs of 2-quasi-symmetric designs

Let a 2-quasi-symmetric design be a $2$-design whose distinct blocks intersect in either $x$ or $y$ points, and consider its point-line incidence graph. The parameter $x$ is the smaller intersection number of distinct blocks.

Nonexistence conjecture. There are no point-line incidence graphs of a $2$-quasi-symmetric design with $x=0$ that have the $M$-property.

This conjecture concerns the classification of distance-regularised graphs with the $M$-property. Existing quasi-symmetric $2$-designs with $x=0$ do not satisfy the relevant condition, but the general nonexistence claim remains open.