We investigate the fraction of close pairs and morphologically identified mergers on and above the star-forming main sequence (MS) at 0.2$leq zleq$2.0. The novelty of our work lies in the use of a non-parametric morphological classification performed on resolved stellar mass maps, reducing the contamination by non-interacting, high-redshift clumpy galaxies. We find that the merger fraction rapidly rises to $geq$70% above the MS, implying that -- already at $z{gtrsim}1$ -- starburst (SB) events ($Delta_{rm MS}geq$0.6) are almost always associated with a major merger (1:1 to 1:6 mass ratio). The majority of interacting galaxies in the SB region are morphologically disturbed, late-stage mergers. Pair fractions show little dependence on MS-offset and pairs are more prevalent than late-stage mergers only in the lower half of the MS. In our sample, major mergers on the MS occur with a roughly equal frequency of $sim$5-10% at all masses ${gtrsim} 10^{10}M_{odot}$. The MS major merger fraction roughly doubles between $z=0.2$ and $z=2$, with morphological mergers driving the overall increase at $z{gtrsim}1$. The differential redshift evolution of interacting pairs and morphologically classified mergers on the MS can be reconciled by evolving observability timescales for both pairs and morphological disturbances. The observed variation of the late-stage merger fraction with $Delta_{rm MS}$ follows the perturbative 2-Star Formation Mode model, where any MS galaxy can experience a continuum of different SFR enhancements. This points to a starburst-merger connection not only for extreme events, but also more moderate bursts which merely scatter galaxies upward within the MS, rather than fully elevating them above it.
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