Let $X\in\{1,2,\ldots,m\}$ be a discrete random variable with $X\sim p$. Let $C$ be a code for $X$ with $l_i$ being the length $i$-th codeword and let $L(C)$ be the expected length of the code. The Shannon code produces a prefix code $C_{Sh}$ with the lengths of the codewords $l_i =\lceil \log_2 \frac{1}{p(i)}\rceil $. The Huffman code produces a prefix code $C_{Hu}$ which is minimal in expected length, but with non-explicit individual codeword lengths. Let $H(X)$ be the entropy of $X$. We have $$H(X)\leq L(C_{Hu}) \leq L(C_{Sh})\leq H(X)+1.$$
My Question: Though Huffman code produces expected lengths at least as low as the Shannon code, are all of it's individual codewords shorter?
Follow-up Question: If not, do the lengths of all the codewords in a Huffman code at least satisfy the inequality: $$ l^{Hu}_i<\log_2 \left(\frac{1}{p_i}\right)+1 ? $$
(I'm looking for proofs/counterexamples.)