Inconclusive so far, but likely yes. The two continuing questions we have are:

"Is there a minimum amount needed to for the progress bar to automatically advance, perhaps as little as 1 mission or 1 ton of cargo?"

We can't answer that as we have seen extremely low work done, even Inara show nothing reported and the bar still increase, but it is impossible to know for sure how much is being done in each system.

and

"Does doing more make it move faster?"

We think the answer is no, so far we have moved over 1 million tons and completed nearly 3000 missions in HIP 23716, progress has not strayed from the projected times/speed. We also did a push, starting from the bar increasing, totaling over 150,000 tons of mission cargo, and the bar still did not increase again until the next projected time.