Let $X$ be a topological space, and $PX$ the space of all paths on $X$. Then let $G\subset X$ be a path-connected subset and $p\in G$ a point. Let $\sigma:G\rightarrow PX$ be a continuous function such that

$\sigma(x)(0)=x\ \forall x\in G,$

$\sigma(x)(1)=p\ \forall x\in G.$

Then show that there exists a function $\tau:G\rightarrow PX$ such that

$\tau(x)(0)=x\ \forall x\in G$

$\tau(x)(1)=p\ \forall x\in G$

$\tau(p)(t)=p\ \forall t\in[0,1].$

Clearly $G$ is contractible, and then the question relates to showing that it is SDR-contractible. I know that this is not always the case (see comb space) but I'm hoping that the addition of the path-connected property makes this true.

I need to show this as part of a lemma in my masters project and would really appreciate any help. Thanks