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We consider exact WKB analysis to a ${\cal PT}$ symmetric quantum mechanicsdefined by the potential, $V(x) = \omega^2 x^2 + g x^2(i x)^{\varepsilon=2}$with $\omega \in {\mathbb R}_{\ge 0}$, $g \in {\mathbb R} _{> 0}$. We inparticular aim to verify a conjecture proposed by Ai-Bender-Sarkar (ABS), thatpertains to a relation between $D$-dimensional ${\cal PT}$-symmetric theoriesand analytic continuation (AC) of Hermitian theories concerning the energyspectrum or Euclidean partition function. For the purpose, we construct energyquantization conditions by exact WKB analysis and write down their transseriessolution by solving the conditions. By performing alien calculus to the energysolutions, we verify validity of the ABS conjecture and seek a possibility ofits alternative form by Borel resummation theory if it is violated. Our resultsclaim that the validity of the ABS conjecture drastically changes depending onwhether $\omega > 0$ or $\omega = 0$: If ${\omega}>0$, then the ABS conjectureis violated when exceeding the semi-classical level, but its alternative formis constructable by Borel resummation theory. The ${\cal PT}$ and the ACenergies are related to each other by a one-parameter Stokes automorphism, anda median resummed form, which corresponds to a formal exact solution, of the ACenergy (resp. ${\cal PT}$ energy) is directly obtained by acting Borelresummation to a transseries solution of the ${\cal PT}$ energy (resp. ACenergy). If $\omega = 0$, then, with respect to the inverse energylevel-expansion, not only perturbative/non-perturbative structures of the${\cal PT}$ and the AC energies but also their perturbative parts do not matchwith each other. These energies are independent solutions, and no alternativeform of the ABS conjecture can be reformulated by Borel resummation theory.

Exact WKB analysis for ${\cal PT}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture