Although the Democrats may have done a dumb thing by initiating an impeachment process, they aren't so stupid as to damage their electoral prospects as well as their credibility in the wake of the abject failure of the Mueller investigation.  So the too-smart-by-half Democratic elites will take the short-term hit to their credibility on an impeachment inquiry rather than vote out articles of impeachment from the floor of the House of Representatives.

First, what kind of articles of impeachment can they write?  The quid pro quo claim is a complete non-starter, where, in essence, they are trying to criminalize the words and the foreign policy of Trump by divining his intention in order to make their case for impeachment.  There is absolutely no law they can point to that he broke in his conversation with President Volodymyr Zelensky.  Second, it can't even claim to be a violation of campaign finance laws. 

So what do the Democrats have to go on?  Absolutely nothing.  Hypothetically, if they were to write up a couple of completely bogus articles of impeachment, they know they don't have a chance of surviving a trial in the Senate.  There is also the possibility that some of the Democrat representatives who were elected in 2018 in districts that Trump won in 2016 will vote against the articles along with some Democrats who actually have a remaining principle or two.

Nancy Pelosi, despite her usually highly partisan public posture, is a shrewd politico who can read the tea leaves, particularly what the polls are indicating about the 2020 election, especially in the battleground states such as Michigan, Wisconsin, and Pennsylvania, as to what will happen if they follow through with impeachment.

Pelosi also realizes that the socialist agenda is a loser with the broader electorate and admitted as much when she recently stated that she's worried about the 2020 candidates being on the "wrong track" and that their agendas, especially those of Warren and Sanders, are losing ones: "I'm not a big fan of Medicare-for-All."  

So she will come up with some kind of excuse, possibly even reverting to her statement last March when she said impeachment must always be a bipartisan affair.  That is not what it is when the votes on establishing rules and guidelines for an inquiry that took place last Thursday were totally partisan with every Democrat (with the exception of two) voting for and every Republican voting against.

There are small signs that even the liberal mainstream media could be setting the table for Pelosi to end the process.  CNN recently ran an opinion piece, "Democrats are Wasting America's Time on Impeachment," where it spelled out in a fairly compelling way why it shouldn't be carried out: 

In the end, Pelosi is doing nothing but checking a box for the most partisan people in her party. There will be nothing to show for it but wasted time and a diversion of the nation's political conversation away from issues that real people care about.

The possibility of Pelosi pulling the plug on impeachment will grow stronger and stronger.  The blowback from the extreme left wing of her party will be deafening, but she knows that to follow through with impeachment will do more damage to the Democrats than to bail out on it altogether.