Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:

we have that $Sec_k(C)$ is normal and $Sing(Sec_k(C)) = Sec_{k-1}(C)$. Does $Sec_k(C)$ have ordinary singularities of multiplicity two along $Sec_{k-1}(C)\setminus Sec_{k-2}(C)$?

More precisely let $f:X\rightarrow\mathbb{P}^n$ be the blow-up of $\mathbb{P}^n$ along $Sec_{k-1}(C)$ with exceptional divisor $E$, and let $Y$ be the strict transform of $Sec_k(C)$. Is the following statement true? The strict transform $Y$ is smooth, it intersects $E$ transversally and we have $$Y = f^{*}Sec_k(C)-2E.$$ I guess this should be true for instance when we consider a rational normal curve $C$ of degree four and $Sec_2(C)$ which is a cubic hypersurface.