Assumption true on specific heat capacity?

Heat conduction reflects how uniform the temperature is within a body, which is controlled by thermal conductivity. Heat capacity is defined as the ratio of input heat to variation in temperature, and reflects a body's capacity for storing heat.

 

Not necessarily. Water has a higher heat capacity than copper but is a much worse conductor of heat.

 

Fluids are more complicated, since they can also transport heat via convection (if they flow). Coming back to original question, heat capacity doesn't play a role in heat conduction, but rather in heat storage.

 

Greenfloss, look up some data on heat conduction and heat capacity.
For simple solids like metal there may be a relationship. Or even with gases.
The theory of heat capacity was important in establishing molecular structure because heat capacity depends on the number of vibrational modes between atoms.

 

I am not aware of a correlation between heat conductivity (W/K/m) and heat capacity (J/kg/K).
However, I would not exclude some correlation for some groups of materials.
For example, it would be good to go back to the kinetic theory of ideal gases to check for such a link.

In the absence of such a link, we have to be more precise about this question.
For example, we could consider heat conduction in materials with the same heat conductivity.
In that case, the heat capacity delays the propagation of heat waves.
It decreases the overall heat conduction.
This is why, in fire protection, materials with high heat capacity might offer some protection, they retard the propagation.
This effect is summarized in a parameter called heat diffusivity, which is essentially the ratio of heat conductivity and heat capacity.