Water on a String
You can indeed conclude that the cohesive force of water is large enough in your experiment to overcome the gravitational force on them.
This is how, even had you started the experiment with the string completely dry and not bothered to tilt the full container towards it, first of all the capillary action would have begun wetting an assumed dry string, and then as it became wet to the extent that the wet bit ended below the level of the water in the full container, only then would the gravity assistance create a siphon of the wet string and the full container would eventually be completely emptied.
I should declare a criterion which legitimises my confirming of your question: I did virtually the same experiment on two occasions years apart and achieved the same result. One time I wetted the string before starting the experiment; the other time I deliberately left it dry. Same result: one progresses to a completely wet string and then the gravity/siphon effect overcomes the capillary tendency to keep the water absorbed in the string.
And that last action I describe especially, as it does relate obliquely to your observation in your own experiment, confirms the veracity of your question, because gravity is demonstrated to be weaker than the cohesive force which is holding together (keeping homogenous?) a continuous column of water in the string, whether stretched tight and (apparently, according to the drawing) not so far from the horizontal in your case or, as in my experiments, let dangle loosely, vertically into the empty container.