@Will Jagy: I've added somthing to my answer to explain why your conjecture is true if "equivalent" means "properly equivalent".

@Will Jagy: It's a good thing you don't recall the proof since there are counterexamples! The principal form $x^2-34y^2$ takes the values $15$ at $(x,y)=(7,1)$ and $-15$ at $(11,2)$ but it does not take the value $-1$, as one can see from its topograph.

The classical argument for this goes as follows. We may assume $f$ is an inclusion by taking its mapping cylinder. The homotopy groups of the homotopy fiber are then the relative groups $\pi_i(Y,X)$. If these are $0$ for $i\leq n$ then $H_i(Y,X)=0$ for $i\leq n$ by the relative Hurewicz theorem. The relative universal coefficient theorem then gives $H^i(Y,X)=0$ for $i\leq n$, so from the long exact sequence we see that $f^*$ is an isomorphism on $H^i$ for $i<n$ and a surjection on $H^n$.

This is a nice argument that I haven't seen before. A small correction: It is $\pi_1(S(E))$ rather than $\pi_1(D(E))$ that acts on $\pi_n(D(E),S(E))$. The map $\pi_1(S(E))\to\pi_1(B)$ is an isomorphism when $n>2$ and a surjection when $n=2$, with kernel acting trivially on $\pi_n(D(E),S(E))=\pi_n(D^n,S^{n-1})$. When $n=1$ the relative $\pi_1$'s are not groups so the argument doesn't seem to work in this case. An orientable line bundle is trivial so one could instead use the K\" unneth formula or a direct argument with exact sequences in homology.

In Definition 1.1 of that paper the dimension of the manifold $W_g$ (which is $2n$, not $n$ as I mistakenly said in my previous comment) is fixed and one is considering embedded submanifolds of ${\mathbb R}^N$ diffeomorphic to $W_g$. Then one lets $N$ go to infinity via the natural inclusion of ${\mathbb R}^N$ in ${\mathbb R}^{N+1}$, so submanifolds of ${\mathbb R}^N$ are regarded as submanifolds of ${\mathbb R}^{N+1}$. However the dimension $2n$ of the submanifolds is not changing. There is no natural way to let $n$ go to infinity in this construction.