Time spent above 0 for time-changed càdlàg functions

Question

Let $\psi \in D([0,T],\mathbb R)$, the Skorokhod space of real-valued càdlàg functions (right continuous with left limits) on $[0,T]$. Let $\Gamma$ be the set of time-changes $\gamma$ that are continuous, increasing, $\gamma(0)=0$ and $\gamma(T)=T$. Is it true that for any $\epsilon>0$ there exists a $\delta>0$ such that if $\sup_{t\in[0,T]}|\gamma(t)-t|\le \delta$, then Lebesgue measure $$ {\rm Leb} \Big( t\in[0,T]: \psi(\gamma(t))<0\le\psi(t) \Big)\le \epsilon\,\,\,? $$ Comment: It is true for $\psi\in C[0,T]$ if ${\rm Leb}(t:\psi=0)=0$, as in that case let $\delta'$ such that ${\rm Leb}(t:|\psi|\le\delta')\le\epsilon$ and by uniform continuity select the $\delta$ so that $\sup_{t\in[0,T]}|\psi(\gamma(t))-\psi(t)|\le \delta'$. Then $$ {\rm Leb} \Big( t\in[0,T]: \psi(\gamma(t))<0\le\psi(t) \Big)\le {\rm Leb} \Big( t\in[0,T]: | \psi(t)|\le \delta' \Big). $$ It is also true for step functions $\psi(t)=\sum_{n=0}^Na_n\mathbf1_{\{[t_n,t_{n+1})\}}(t)$, as if $\sup_{t\in[0,T]}|\gamma(t)-t|=\sup_{t\in[0,T]}|t-\gamma^{-1}(t)|\le \epsilon/(N-1)$, using $$ {\rm Leb} \Big( t\in[0,T]: \psi(\gamma(t))<0\le\psi(t) \Big)\le {\rm Leb} \Big( t\in[0,T]: \psi(\gamma(t))\neq \psi(t) \Big), $$ and \begin{equation}\begin{split} \psi(t)-\psi(\gamma(t))%&=\sum_{n=0}^Na_n\mathbf1_{\{[t_n,t_{n+1})\}}(t)-\sum_{n=0}^Na_n\mathbf1_{\{[t_n,t_{n+1})\}}(\gamma(t)) \\ &=\sum_{n=0}^Na_n\mathbf1_{\{[t_n,t_{n+1})\}}(t)-\sum_{n=0}^Na_n\mathbf1_{\{[\gamma^{-1}(t_n),\gamma^{-1}(t_{n+1}))\}}(t)\\ &=\sum_{n=0}^Na_n\left(\mathbf1_{\{[t_n,t_{n+1})\}}(t)-\mathbf1_{\{[\gamma^{-1}(t_n),\gamma^{-1}(t_{n+1}))\}}(t)\right) \end{split}\end{equation} we deduce that $(t:\psi(t)-\psi(\gamma(t))\neq 0)\subset \cup_1^{N-1} [\gamma^{-1}(t_n)\wedge t_n,\gamma^{-1}(t_n)\vee t_n]$, which Lebesgue measure is less than $\epsilon$. Of course this proof is of no use even for continuous $\psi$.