Let's 1st understand what these planets represent on their own?

Sun - Sun represents Soul, Personality, King, Government, Father, Authority, Career, Top Positions, Ego, Self Esteem, Health, Education etc.

Moon - Moon represents Mind, Emotions, Mother, Local Government, Peace of Mind, Home Environment, Water, Milk etc.

Ketu - Just like Rahu, Ketu also represents Illusion as both are ilusionary planets but the difference is that Ketu is also the one who removes the illusion and brings enlightenment. Besides this, Ketu represents Isolation, Separation, Spirituality, Mystical World, Other Dimension means things related to Alien World, also represents Son, Dog, Intuition, Research Ability, Curious Nature, things which we have already achieved in past life and not much concerned to perform in this life. 

3 planetary conjunction - Whenever 3 or more planets are conjunct with each other in a house/sign then things related to that house/sign become a kind of life focus for person. It is because of huge accumulation of energy within 30 degrees of sky. It means not only these 3 planets' energy is going into this house/sign but the energy of at least 4 to 6 house/sign is going into this house/sign as these planets will be ruling some houses in chart. So, this house/sign with conjunction becomes almost the most important house in chart. During Career Consultation, I give a huge importance to any such house which has 3 or more planets in it.

House Ruler - Besides these karaka things, Sun, Moon & Ketu will also rule some houses in your chart and will represent things related to that house. Of course we can't cover conjunctions Ascendant and house wise else it will never be over. So, you have to include things related to the houses at the time of interpreting this conjunction. Also, we need to keep in mind all Yogas and Raj-Yogas which their conjunction would be making to completely understand the conjunction. Here, we can only cover meaning as per karakas.

Meaning of Sun-Moon-Ketu Conjunction - We have yet another stressful conjunction here. This is a stressful conjunction for both Sun & Moon as they are conjunct with their enemy Ketu. If Moon is in between Sun & Rahu degree-wise, then it creates a kind of Paap Kartari Yoga as gentle Moon would be sitting in between 2 strong malefics. In such cases, mind will take lots of stress and can be termed as depressed mind. This is another conjunction in a birth-chart which shows that person was born during eclipse month and close to lunar eclipse. As Sun-Moon both are impacted by Ketu, this may lead to another situation where person is separated from parents or he feels a natural disconnect from them. Ketu with Sun-Moon shows that person is isolated from himself, i.e. he doesn't have a right idea about himself or his right path in life. This conjunction also leads person towards occult, mysticism and spiritual world as Ketu doesn't feel satisfied with material success or wealth. Ketu with Sun-Moon certainly makes someone feel that he doesn't belong to this place called Earth. Moon represents Mind and Ketu is other worldly stuff like occult, mystical or spiritual by nature. So, person finds peace of mind in mystical and spiritual world. Person's mind gets an easy access to other worldly stuff. Ketu's association with Sun gives no sense of self or confidence. Sooner or later this person draws himself into mystical world and finds his peace an confidence there. This person may not like working under others authority. If Sun & Moon are in good dignity then only difference can be that parents may be supportive of his spiritual path of life. It won't change Ketu's nature a bit. 

Who will win the Conjunction? - Ketu, as it is impacting Sun & Moon both. 

Dignity & Strength - 

In any sign, Ketu will play its role as stated above. So, dignity doesn't really matter here.