= Solution
For a positive <electron neutrino> matter potential and the normal ordering implicit in positive $K$, choose $0\le\theta_m\le\pi/2$ by continuous diagonalization of the matter <Hamiltonian>. With $D=[(\cos2\theta-K\rho)^2+\sin^22\theta]^{1/2}$, the quadrant is fixed by
$$
\cos2\theta_m=\frac{\cos2\theta-K\rho}{D},\qquad\sin2\theta_m=\frac{\sin2\theta}{D}.
$$
At high density $\cos2\theta_m\to-1$, so
$$
\boxed{\theta_m\to\pi/2,\qquad \lvert\nu_e\rangle\simeq\lvert\nu_{2m}\rangle.}
$$
Taking a principal-value arctangent alone and concluding $\theta_m\to0$ would select the wrong branch. The production state is the second \i[matter propagation eigenstate], rather than literally the vacuum <mass eigenstate> while still inside the dense medium.
Assume the solar density decreases slowly enough for <adiabatic matter-enhanced neutrino conversion>, and neglect subsequent Earth matter effects. The state follows $\nu_{2m}$ continuously into the vacuum <mass eigenstate> $\nu_2$. Since $\langle\nu_e\vert\nu_2\rangle=\sin\theta$,
$$
\boxed{P_{ee}^{\rm adiabatic}=\sin^2\theta=\sin^2(22.5^\circ)=\frac{2-\sqrt2}{4}\simeq0.1464.}
$$
This is substantially below the phase-averaged vacuum result $0.75$. There is no interference between two populated vacuum <mass eigenstates> in this ideal adiabatic limit. More generally, after phase averaging, $P_{ee}=\tfrac12+\tfrac12\cos2\theta\cos2\theta_m$ at production if the evolution is adiabatic. In the high-density limit, a nonadiabatic level-jump <probability> $P_c$ instead gives $P_{ee}=(1-P_c)\sin^2\theta+P_c\cos^2\theta$. Thus the supplied core-density condition alone does not guarantee the numerical Earth result: the usual solar adiabatic assumption is also needed. These are the stipulated two-flavor parameters, not a precision prediction for the measured three-flavor solar spectrum.
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