Let $D_n$ be an $n$-disc. Is the factor map $p: D_n\to D_n/S^{n-1}\simeq S^n$ a Serre fibration, in other words, can any homotopy $F: [0,1]\times X\to S^n$ be lifted to $\tilde{F}: [0,1]\times X\to D^n$, if a lift $\tilde{f}_0$ of $F(0,\cdot)$ is given?
More generally, if $p(S^{n-1})=\{s_0\}$, can a homotopy $F: [0,1]\times (X,A)\to (S^n, s_0)$ be lifted to $[0,1]\times (X,A)\to (D_n, S^{n-1})$, if $\tilde{f}_0$ is given? Suppose that $X$ is nice, such as a CW or simplicial complex.