It's still fuzzy just what the tension is, but this article seems to be saying that at the time of the CMBR (13.8 Gyrs ago) the expansion rate was slower. But the tension isn't comparing today's rate with the rate at the time of the CMBR, but, rather, the extrapolated rate of the CMBR. IOW, the theoretical rate that we should have today, using assumptions like DE, is being compared with the observed expansion rate. Is this the "tension"?
But can we accurately model today the historic DE effect on the cosmos, at least from our point of view? Then there's the likelihood that DE isn't an acceleration, but and acceleration rate of acceleration (jerk) since DE keeps getting strong with time. Isn't this right?
The other main force determining the rate, of course, is the gravitational effect of spacetime. So, yes, our large "void" would have a little less pull-back by gravity, thus DE would gain even more speed. But is this difference adequate to explain the tension?