The first-increase inequalities for connected unit interval graphs

Let $G$) be a connected unit interval graph defined by the weakly increasing sequence ${\bf m}=(m_1,m_2,\ldots,m_{n-1})$, where $i\leq m_i\leq n$ for all $i\in[n-1]$. Let $m_r<m_{r+1}$ be the first increase, and let $G'$ be the unit interval graph defined by

$${\bf m}'=(m_1+1,\ldots,m_r+1,m_{r+1},\ldots,m_{n-1}).$$

First-increase inequalities. One has

$$G\geq_e G'\qquad\text{and}\qquad G\geq_s G'.$$

These inequalities are proposed as part of a sequence of comparisons between connected unit interval graphs that could establish their $e$-positivity and Schur-positivity. The Schur-positivity consequence is already known for unit interval graphs, while the corresponding $e$-positivity goal remains open in the stated context.