Is modality-specific working memory capacity correlated within individuals? The multi-component model of working memory posits different processes for phonological and visual working memory storage. 

Working memory capacity can be measured in a variety of ways (see: How to reliably measure working memory capacity?), and we know that there is reliable individual variation on working memory capacity. 

Is there a strong correlation of working memory capacity in different modalities within an individual? Put another way: If I have measured working memory capacity for an individual using a visual working memory task (like a Matrix span task), how reliably can I predict performance on a verbal working memory task for that individual?
How working memory breaks down, or even whether it is a valid construct at all, is still somewhat controversial.  The evidence for domain-specific modalities is largely based on the (lack of otherwise expected) interference between them.  Another line of evidence that could be used to validate this model is checking the correlations between them: A high correlation would argue for a domain-general working memory, while a low correlation would support domain-specific modalities.  In practice, however, the value of examining correlations between working memory measures is confounded by a general capacity factor (eg, executive control), interaction with short-term memory, the possibility of any number of additional modalities (for example, visual vs. spatial), and the result of many years of assuming a unitary working memory construct resulting in a variety of measures that are validated against each other (ie, expected to correlate highly).

Nonetheless, just such a test was done by Alloway, Gathercole &amp; Pickering (2006), who subjected over 700 primary school children to a battery of different working memory capacity measures.  As expected, correlations between many of the measures were statistically significant, but importantly, not commensurate with either a simple domain-specific model, or a simple general model with WM/STM split.  To answer your question, here is a table from the paper that summarizes some of the results (more detail here):


As you can see, the correlations, while nearly all significant at p>.001, vary considerably.  These results are consistent with similar studies that review several independent measures of working memory that nonetheless all tend to correlate well with cognitive factors such as general intelligence.