Maximum common-prefix conjecture for uniform squarefree morphisms

Let $T$ be the alphabet, and let an $n$-uniform squarefree morphism be a morphism $h:T^*\to T^*$ for which every image $h(a)$ has length $n$ and which maps squarefree words to squarefree words. Let $\operatorname{lcp}(h)$ be the length of the longest common prefix of the images $\{h(a):a\in T\}$, and let $\operatorname{lcp}(n)$ be the maximum value of $\operatorname{lcp}(h)$ over all $n$-uniform squarefree morphisms $h:T^*\to T^*$. Maximum common-prefix conjecture. For every $n\geq 30$, there exists an $n$-uniform squarefree morphism $h:T^*\to T^*$ such that $\operatorname{lcp}(h)=n-6$. The computed values for $18\leq n\leq 70$ suggest this conjecture; its validity for all $n\geq 30$ remains open.