Thank you gentlemen for your replies,
Bailey: there wasn't any black gummy material in the hose between the fill valve and the tub, so it's not this one.
keithvibe: there actually was an excess soap and fabric softner build up between the tubs, which I cleaned using the cloth soaked in the vinegar. We just tried small and then medium wash and so far so good. The problem came during a large wash, so we should try it at some point. As for now I'm trying to understand how that build up could cause grease particles getting into the tab. Is there any sound explanation to it?
Thank you so much,