Aligning pixels for using GSW as mask layer with Landsat 8 and Global Surface Water (JRC) Layer

Question

I want to isolate some surface water areas on Landsat 8 images by using the Global Surface Water dataset in GEE as mask. After mapping both in the Editor I realized the positions of pixels of GSW and Landsat8 do not align, instead there is an offset in both dimensions (see image).

I did simply load and display the data. Didn't reproject any, as GEE states to harmonize Layers automatically. Looking in the properties of the images there are different projections assigned to them (LS: EPSG:32624 and GSW: EPSG:4326).

Is it necessary to reproject one of the layers?

I did try the function .register(), which should find displacements and correct them. However the output layer didn't have any changes in pixel position.

var LC08_CE1_2019= ee.Image('LANDSAT/LC08/C01/T1_SR/LC08_217063_20190819');
var gsw = ee.Image('JRC/GSW1_0/GlobalSurfaceWater');
var extentGSW = gsw.select('max_extent') 
  .clip(Ceara)
  .unmask(0);
var regGSW= extentGSW.register({referenceImage: LC08_CE1_2019.select('B1'), maxOffset:15.0, patchWidth:60.0, stiffness:10});

Answer 1

Since the question has been updated i'll repost the answer from comments. But you can use reproject.

var LC08_CE1_2019= ee.Image('LANDSAT/LC08/C01/T1_SR/LC08_217063_20190819');
var gsw = ee.Image('JRC/GSW1_0/GlobalSurfaceWater');
var extentGSW = gsw.select('max_extent') 
  .clip(geometry)
  .unmask(0);
var regGSW= extentGSW.register({referenceImage: LC08_CE1_2019.select('B1'), maxOffset:15.0, patchWidth:60.0, stiffness:10});

ee.Image('LANDSAT/LC08/C01/T1_SR/LC08_217063_20190819'); 
Map.addLayer(LC08_CE1_2019) 
Map.centerObject(LC08_CE1_2019, 8) 
var gsw = ee.Image('JRC/GSW1_0/GlobalSurfaceWater'); 
var extentGSW = gsw.select('max_extent').clip(LC08_CE1_2019.geometry()).unmask(0); 
Map.addLayer(gsw) 
var reproject = gsw.reproject(LC08_CE1_2019.projection()) 
Map.addLayer(reproject)