I am not a fan of the SLS. So take my comments with that as a grain of salt.
SLS is fundamentally a jobs program. It has no mandate to be affordable. It also has no real mission at the moment. Thus there is nothing really pushing its development other than jobs in states of politicians.
Thus logic and rationality do not apply to SLS. It will be cancelled when the politicians stop fighting for it.
Now on a technical level, Falcon Heavy is pretty powerful, but it is not actually powerful enough to replace SLS, phase 1 or phase 2. Falcon Heavy is about 54 Tonnes (Metric Tonne) to LEO.
Crossfeed, which was initially discussed has been discarded as the performance of the base Falcon 9 core improved. The first Falcon Heavy will launch with Block 3 side boosters. Unclear if they built a Block 5 center core, but it is a slightly different booster as the center core. Once they transition to all being Block 5 cores, performance is expected to hit their old numbers without needing crossfeed. It also turned out to be way harder than they expected, and then became unnecessary and they pivoted.
SLS is 70T (Block 1) going up to 130T (Block 2). That is a big differential and in this space, it makes a big difference.
But if you can launch a Falcon Heavy, fully reusable for the prices SpaceX is quoting (\$90-\$125 million), several times a year, vs SLS once every 2-4 years for billions of dollars a launch, the choice seems obvious. Build smaller payloads in pieces and actually get your payload launched. Cadence matters almost as much as payload.
As currently designed, although the Falcon Heavy has a 54 mT payload, the actual payload adapter that SpaceX produces is limited to much less mass. It is unlikely in the near future we will see any LEO boosts with the full payload mass, but if a customer wants to I am sure SpaceX will modify the payload adapter.
On the other hand, the fairing on Falcon Heavy is the same as the Falcon 9 fairing and is in fact, possibly one of the most limiting factors. It is a 3m core, with a hammerhead fairing, extending about a meter on either side, for a 5m wide fairing. Alas, there is 'stuff' on the inside that reduces the size enough to differentiate from the Atlas booster's 5m fairing which has more internal space. SpaceX faces this issue with launching their 4400 satellite Starlink constellation, where they find the booster can launch the needed mass, but the fairing cannot hold enough satellites.
SLS will have MUCH larger fairings around 8.5 to 10 meters wide, for which there may certainly be good use cases. However you have to trade off the cost, low flight rate, against building the payload in smaller pieces.
One of SLS's big problems is lack of a mission. Thus they need to find something really important to do with SLS. A realist who wants their mission launched would be wiser to consider Falcon Heavy with multiple launches and in space assembly to actually get launched.
The bigger telltale will be when you see a push to launch Orion on Falcon Heavy. That will be the death knell for SLS.
My personal prediction is SLS will actually launch (late), at least twice, which takes us out to at least 2020, but not too many times more. Thus to the specific question asked, the answer would be no.
BFR/BFS from SpaceX will be the SLS killer, if they succeed at developing it. That looks to be available in the same time frame, and is more likely the SLS killer.
A commenter notes that if the Grey Dragon mission happens, two paying customers, on a loop around the moon, (Almost Apollo 8, but not actually entering Lunar orbit, rather a simple free return mission), to expect SLS to be cancelled. I am not so sure, but it would certainly make supporting SLS that much harder.