Lehmann, in Theory of Point Estimation p.212, defines scale median as the solution to: $${E(X)I(X\le c)} = {E(X)I(X\ge c)}$$
given $X$ is a positive random variable, and ${E(X)}< \infty$.
Now based on this: $${E(X)I(X\le c)} = \int_0^c x\,f(x)\,dx \le c $$ and thus conversely, $${E(X)I(X\ge c)} = \int_c^\infty x\,f(x)\,dx \ge c $$
Therefore, the equality will never hold unless $X$ is a discrete point at $c$ w.p. 1.
So what's wrong with my reasoning?
"Conversely" is wrong and misses the fact that one uses two inequalities to get an upper bound of $\mathbb E(X;X\leqslant c)$, namely:
(1) If $x\leqslant c$, then $xf(x)\leqslant cf(x)$.
(2) $\displaystyle\int_{-\infty}^cf(x)\mathrm dx\leqslant1$.
This approach fails with $\mathbb E(X;X\geqslant c)$ because the inequality in (1) is reversed while the inequality in (2) is not, to wit:
(1') If $x\geqslant c$, then $xf(x)\geqslant cf(x)$.
(2') $\displaystyle\int_c^{+\infty}f(x)\mathrm dx\leqslant1$.