WASHINGTON (CNN) As Robert Mueller's investigation into Russian interference in the 2016 election inches closer to the President, approval of Donald Trump's handling of the investigation has fallen, matching its all-time low in CNN polling. The findings, from a new CNN poll conducted by SSRS , come as half of Americans say they think it is likely that the Mueller investigation will implicate the president in wrongdoing.

In the new poll, Trump's approval rating for handling the Russia investigation dips to 29%, matching a low previously hit in June of this year.

The survey was conducted from Thursday through Sunday, a time when court filings in cases against Paul Manafort and Michael Cohen revealed the alleged lies Donald Trump's former lawyer and former campaign chairman told either publicly or to the special counsel's investigators. Trump claimed over the weekend that the filings clear him of any wrongdoing and called for the investigation to end. But the Cohen filing implicates Trump in the hush-money scheme to pay women alleging affairs with Trump so they would stay silent during the campaign, and the Manafort filing suggests the former campaign chair continued to lie about contacts with White House this year.

But Mueller's approval rating is also down in the poll: 43% approve and 40% disapprove. That compares to a 48% approve to 36% disapprove split in early October. The dip in Mueller's numbers comes almost entirely among independents, among whom approval has fallen 10 points to 36%. Among partisans on both sides, Mueller's approval holds about even with where it was in an October survey -- 71% of Democrats approve as do 21% of Republicans.

Trump's approval rating drop, however, comes among his own partisans as well as among independents. Among Republicans, 51% approve of Trump's handling of the investigation, a new low by one point, while among independents, 26% approve, also a new low. Just 15% of Democrats approve of the president's handling of the investigation, up from October but about on par with the level who felt that way earlier this year.

Overall, a majority (54%) continue to say that most of the things Trump has said publicly about the Russia investigation are false, while just over a third say they are mostly true (36%). That's largely unchanged since August.

There has also been no meaningful change on whether the investigation itself is a serious matter or mainly an effort to discredit Trump's presidency: 59% say it's a serious matter, 35% an effort to discredit Trump.

Half of Americans think it is very or somewhat likely that the Mueller investigation will implicate Trump personally in wrongdoing. That figure is higher among Democrats (78% say it's likely), but still, nearly a quarter of Republicans think Trump is likely to be personally implicated (23%) as do about half of independents (47%). Aside from partisanship there's a stark divide here by education among whites, with 58% of whites with college degrees saying they think Trump is likely to be implicated vs. 43% of whites without degrees.

Looking at Michael Cohen's recent revelation that work continued on a potential project in Russia during the 2016 campaign, 44% believe Trump acted unethically in considering projects in Russia during the campaign, 26% say it was unwise but not unethical, and 23% say there was nothing wrong with Trump's action.

Trump's overall approval rating for handling the presidency matches its pre-election level just about exactly, 39% approve and 52% disapprove. Trump's favorability rating is also steady at 40% favorable to 55% unfavorable.

The president's approval rating for handling the economy has dipped (49% now, was 53% in early November), but it holds about even on the rest of the issues tested in the poll. Around 4 in 10 approve of the way the president is handling immigration (39% approve), foreign trade (39% approve) and foreign affairs (36% approve). Fewer, 31%, approve of the way the president is handling environmental policy.

Republicans are generally more positive about the president's performance on issues other than the Russia investigation. More than 8 in 10 approve of the way he's handling the presidency generally, the economy and immigration, more than three-quarters of Republicans approve of his handling of foreign affairs and foreign trade, and 69% approve of his work on environmental policy.

The CNN Poll was conducted by SSRS December 6 through 9 among a random national sample of 1,015 adults reached on landlines or cellphones by a live interviewer. Results for the full sample have a margin of sampling error of plus or minus 3.8 percentage points, it is larger for subgroups.