Non-isogenous elliptic curves (Zoom)

Let $E_f: y^2=f(x)$ and $E_h: y^2=h(x)$ be elliptic curves over a field $K$ of characteristic zero that are defined by cubic polynomials $f(x)$ and $h(x)$ with coefficients in $K$.
Suppose that one of the polynomials is irreducible and the other reducible. We prove that if $E_f$ and $E_h$ are isogenous over an algebraic closure $\bar{K}$ of $K$ then they both are isogenous over $\bar{K}$ to the elliptic curve
$$y^2=x^3-1.$$

References.
[1] Yu. G. Zarhin, Homomorphisms of hyperelliptic jacobians. Trudy Math. Inst. Steklova 241 (2003), 79–92; Proc. Steklov Institute of Mathematics 241 (2003), 90–104.
[2] Yu. G. Zarhin, Non-isogenous superelliptic jacobians. Math. Z. 253 (2006), 537–554.
[3] Yu. G. Zarhin, Non-isogenous elliptic curves and hyperelliptic jacobians. Math Research Letters, to appear; arXiv:2105.03783 [math.NT].
[4] Yu. G. Zarhin, Non-isogenous elliptic curves and hyperelliptic jacobians. II. MPIM preprint series 29 (2022); arXiv:2204.10567 [math.NT].