Order-4 meridian and stable irreducibility conjecture for the Klein bottles $\mathcal K(l,m,n)$

Let $\mathcal K(l,m,n)$ denote the Klein bottle constructed from the parameters $l,m,n$. Its meridians are said to have order 4 when they represent elements of order $4$ in the fundamental group of the complement; it is indecomposable if it cannot be expressed as a nontrivial connected sum, and stably irreducible if it does not become reducible after stabilization. Order-4 meridian and stable irreducibility conjecture. Each $\mathcal K(l,m,n)$ has order-4 meridians, is indecomposable, and is stably irreducible. The preceding proposition shows that each such Klein bottle is either stably irreducible or stably isotopic to a projective plane with order-4 meridians; the conjecture asserts the stronger alternative and also includes indecomposability.