Origins of the interdecadal fluctuations of the ENSO-Indian summer monsoon (ISM) teleconnection strength measured by running window correlation coefficients have been much debated. The main question is whether it’s due mostly (i) to sampling errors, or, (ii) under internal variability decadal time scale drivers modulate it, and/or, (iii) it has undergone some forced change recently. A new statistical test does not detect any of (ii-iii) in observations. However, it does not mean that there could not be such effects, just that the data is insufficient. Large ensemble data sets are well suited to investigate these questions. Previously, (iii) was found to be small (even if possibly important) in the MPI-GE data set, and the situation is similar in the new CESM2-LE data set. This time we investigate (ii), too, and find that the decadal variability of e.g. the Dipole Mode Index (DMI*, decorrelated from Nino3) of the Indian Ocean “explains” 4% of the apparent variability of the ENSO-ISM teleconnection. If the CESM2-LE were faithful to reality, then this signal would take two millenia to have a 50% chance to detect. We also find that this apparent DMI* influence is due to heteroscedasticity, namely, to the apparent DMI* influence on ENSO variance. However, we also find that this influence is mostly apparent, indeed, as it is the ENSO variance that dominantly influences the DMI*.