On the questions P ?= NP $\cap$ co-NP and NP ?= co-NP for infinite time Turing machines

Schindler recently addressed two versions of the question P $$ NP for Turing machines running in transfinite ordinal time. These versions differ in their definition of input length. The corresponding complexity classes are labelled P, NP and ${\mathrm{P}}^{+},{\mathrm{N}\mathrm{P}}^{+}$. Schindler showed that P $\mathrm{\ne }$ NP and ${\mathrm{P}}^{+}\mathrm{\ne }{\mathrm{N}\mathrm{P}}^{+}$. We show that P $=$ NP $\cap$ co-NP and NP $\mathrm{\ne }$ co-NP, whereas ${\mathrm{P}}^{+}\subset$ NP $\cap$ co-NP and ${\rm NP}^+ \neq $co-NP$^+$.