Immersing 2-torsion lens spaces

Let $L^{2n+1}(2^r)$ denote the mod $2^r$ lens space, and let $\alpha(n)$ denote the number of 1's in the binary expansion of $n$. In two recent papers, J. González has obtained nonimmersion results, of different types depending upon whether or not $r \geq \alpha(n)$, for $L^{2n+1}(2^r)$ in Euclidean space. He conjectures that, for suitable values of $n$ and for $r$ large enough, $L^{2n+1}(2^r)$ does not immerse in $\mathbb{R}^{4n-2 \alpha(n) +2}$. I will elaborate on the preceding and discuss results that compare nicely with González's work.