Partial converse to Proposition 3(1): goodness and the identity section

Let $B$ be a compactly generated space such that the image of every continuous map $S\to B$ from any compact Hausdorff space $S$ in the same universe is closed in $B$. Let $G$ be a simplicial parametrized group over $B$: it assigns a parametrized group to every object of the opposite simplex category and continuous maps over $B$ preserving units and multiplication to its morphisms, with pointwise identity and composition laws. Denote its degree-$n$ total space by $E_n$, its projection by $\pi_n:E_n\to B$, and its unit section by $e_n:B\to E_n$; each $E_n$ is compactly generated, inversion is continuous over $B$, and multiplication is continuous on the compactly generated replacement of the pairs in the same fiber and satisfies the group laws there. To state the hypotheses and conclusion explicitly, write $I=[0,1]$, and let $k(T)$ retain the set of $T$ with the finest topology making all originally continuous maps from compact Hausdorff spaces in the same universe continuous. Maps over $B$ are continuous maps commuting with projections; a homotopy over $B$ from $f_0:S\to T$ to $f_1:S\to T$ is a continuous $F:k(S\times I)\to T$ with $F(s,0)=f_0(s)$, $F(s,1)=f_1(s)$, and $\pi_T(F(s,t))=\pi_S(s)$. For a map $r:P\to Q$ over $B$, let $\mathcal C(r)$ mean the following lifting property: for all compactly generated spaces $U,V$ over $B$ and maps $p:U\to V$ over $B$ satisfying both conditions below, all maps $a:P\to U$ and $b:Q\to V$ over $B$ with $p\circ a=b\circ r$ admit a map $\ell:Q\to U$ over $B$ with $\ell\circ r=a$ and $p\circ\ell=b$. The first condition on $p$ is that, for every compactly generated space $Z$ over $B$, map $c:Z\to U$ over $B$, and continuous $H:k(Z\times I)\to V$ with $\pi_V(H(z,t))=\pi_Z(z)$ and $H(z,0)=p(c(z))$, there exists a continuous $L:k(Z\times I)\to U$ with $\pi_U(L(z,t))=\pi_Z(z)$, $L(z,0)=c(z)$, and $p(L(z,t))=H(z,t)$ for all $z,t$. The second condition is the existence of a map $q:V\to U$ over $B$ and homotopies over $B$ from $q\circ p$ to $\mathrm{id}_U$ and from $p\circ q$ to $\mathrm{id}_V$. Assume $\mathcal C(s_i:E_n\to E_{n+1})$ for every $n\geq0$ and every $0\leq i\leq n$, where $s_i$ is assigned to the opposite of the simplex surjection identifying $i$ and $i+1$, and assume $\mathcal C(e_0:B\to E_0)$, where $B$ has its identity projection. Then $\mathcal C(e_n:B\to E_n)$ holds for every $n\geq0$, including $n=0$. No nonemptiness assumptions are imposed on the spaces.