Even alternating-deletion count for a noninjective label sequence

Write $[a]=\{0,\ldots,a-1\}$ for $a\in\mathbb N$ (empty when $a=0$). A signed label on $[m]$ is a pair $(\varepsilon,j)$ with $\varepsilon\in\{+,-\}$ and $j\in[m]$; negation reverses its sign. For a sequence $M:[\ell]\to\{+,-\}\times[m]$, write $\operatorname{Alt}_+(M)$ when there exists a strictly increasing $u:[\ell]\to[m]$ such that $\operatorname{im}M=\{((-1)^a,u(a)):a\in[\ell]\}$, and define $\operatorname{Alt}_-(M)$ by reversing all these signs. Here $(-1)^a$ denotes the positive sign for even $a$. These predicates concern the label set ordered by index, not the input order. Let $k,m\in\mathbb N$ and $L:[k+1]\to\{+,-\}\times[m]$. Assume $L$ is not injective and $L(i)\ne-L(j)$ for all $i,j\in[k+1]$. Let $L^{\widehat i}$ be the sequence obtained by deleting position $i$ and retaining the order of the other entries. Then