In metal carbonyls, bonding involves both $\sigma$ donation from the carbonyl carbon to the metal and $\pi$ back-donation from the filled $d$ orbitals of the metal into the vacant antibonding $\pi^*$ orbital of carbon monoxide. This back-donation strengthens the metal-carbon (M-C) bond but weakens the carbon-oxygen (C-O) bond. A higher positive charge on the central metal ion reduces its ability to back-donate electron density to the CO ligand. Among the given complexes, Mn is in the $+1$ oxidation state in $[Mn(CO)_6]^+$, while Fe and Cr are in the $0$ oxidation state, and V is in the $-1$ oxidation state. Because Mn has the highest positive charge, the extent of synergic back-bonding is the least, making the C-O bond the strongest in $[Mn(CO)_6]^+$.