The honeycomb lattice iridates ${A}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$ ($A=\mathrm{Na}$, Li) are spin-orbit assisted Mott insulators proximate to Kitaev's quantum spin liquid. The insulating state as well as the magnetic properties are believed to arise due to a delicate balance of several energy scales. We report on high-pressure electrical transport and x-ray-diffraction measurements on ${A}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$ ($A=\mathrm{Na}$, Li) in an attempt to study their structural and electronic evolution with pressure. We found that while $\mathrm{L}{\mathrm{i}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$ undergoes a structural phase transition into the dimerized state at a pressure of $P$ \ensuremath{\sim} 4 GPa, in $\mathrm{N}{\mathrm{a}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$ the conservation of the original $C2/m$ structure up to at least 58 GPa is observed. In addition, $\mathrm{L}{\mathrm{i}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$ undergoes a sluggish structural rearrangement at the pressure range 20--40 GPa coinciding with a significant decrease in resistance. Despite dissimilar structural evolution and different mechanisms of the electrical conductivity, Arrhenius conductivity for $\mathrm{N}{\mathrm{a}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$ and Mott variable-range hopping in $\mathrm{L}{\mathrm{i}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$, both systems show a very similar $R$($P$) behavior. Namely, after a nonmonotonic decrease of the resistance $R$ and the charge gap \ensuremath{\Delta}, the \ensuremath{\Delta} stabilizes at about 45 GPa and even increases slightly with pressure; the $R$($T$) shows insulating behavior up to the highest pressure measured, 80 and 55 GPa, respectively. This resilient nonmetallic behavior of the studied iridates suggests a formation close to a localized-itinerant crossover of unusual electronic states, whose possible features are discussed. Unforeseeably, the $R$($P$) behavior is not dependent on the buffer element $A$, which seems essential for understanding the nature of the electrical conductivity in iridates.
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